Quantum error detection ($PEP=c_{E}P$)

A code with orthogonal projector $P$ detects an error $E$ when its compression to the code is scalar. A projective measurement of $P$ then either flags departure from the code or, on a successful code-space outcome, leaves the logical state unchanged up to normalization. This includes harmless scalar errors, even though they need not produce a nonzero error syndrome. Detectability extends by linearity to the span of a set of errors. Every Pauli operator of weight less than the quantum code distance is detectable.

Split the support of a Pauli operator of weight at most $2t$ into two sets of size at most $t$. It is, up to a scalar phase, $E_a^{†}{E}_b$ for two errors in the correctable family. The Knill--Laflamme condition makes its compression to the code scalar, which is quantum error detection. Expansion in the Pauli basis extends the assertion to every operator supported on at most $2t$ qubits. The number $t$ counts affected qubits, not the cardinality of an arbitrary selected family of errors.